What are the three ways to solve a quadratic equation?
- Completing the square
- The quadratic formula
What are the steps to solving a quadratic equation by factorising?
- Get everything on one side, leaving just 0 on the other. (i.e. x2-2x+1=0).
- Factorise, so you get something like (x+a)(x+b)=0.
- Solve each parenthesis for x. (i.e. x+a=0 and x+b=0)
How can you recognize that an equation is quadratic?
Some variable is squared.
It can always be written as:
What are the steps for factorising a quadratic equation?
- Factor out any common factors. (So anything that goes into all of the terms. i.e. 6x2-2x-8 = 2 (3x2-x-4))
- Make a list of all the factors that multiply to make ac.
- Check to see which two factors from step two add to the b term.
- Rewrite the equation, separating the middle term into the two factors you found.
- Take the common factors out of the first two terms and the last two, and regroup.
For example, 3x2-x-4:
- 6x2-2x+8=2(3x2-x-4). Now I need to factorise 3x2-x-4
- 3•-4=-12, so the factors are 1•-12, -1•12, 2•-6, -2•6, 3•-4, -3•4
- Do any of the pairs add to -1? Yep, 3+-4=-1.
- 3x(x+1) - 4(x+1) = (3x-4)(x+1). So the answer is 6x2-2x+8=2(3x-4)(x+1).
What does the graph of f(x)=x2 look like? What are the vertex and line of symmetry?
Line of symmetry: x=0
How do you shift f(x)=x2 to the right by h?
You put a "-h" in with the x term.
How do you shift f(x)=x2 to the up by k?
You put a "+k" at the end.
How do you mirror f(x)=x2+k over the x-axis?
You multiply the whole function by -1.
How do you mirror f(x)=x2+k over the y-axis?
You multiply the x-part by -1.
How do you vertically stretch f(x)=x2+k by a?
You multiply the whole function by a.
How do you horizontally stretch f(x)=(x-h)2+k by b?
You multiply the whole x-part by a.
Where is the vertex and line of symmetry for
Vertex: (h, a•k)
Line of symmetry: x=h
How can you find the root if a quadratic equation has "two equal real roots"?
This means the discriminant is zero, so your solution is just